Let \(G\) be a game in extensive form
A subgame of \(G\) is
Recall perfect-information chicken:
I propose that \(\{S, (SG, SG)\}\) and \(\{R, (RD, RD)\}\) are both bad predictions. Why?
Barry gets to see Andrea’s move before making their own
Upon seeing Remain, Barry would never Stand Ground
Upon seeing Swerve, Barry has no reason to Roll Dodge
Barry’s optimal action depends on which subgame he is in
At \(\{R, (RD, SG)\}\), Barry receives a payoff of 0
How could Barry improve his payoff?
He could threaten to always Stand his Ground
Imagine that Barry can FlexSeal his feet to the pavement, and Andrea sees him do this while driving
Recall: in intermediate micro (EC 311), individual agents maximize their expected payoffs subject to some constraints
If the agent can choose \(x\) and \(y\) to get utility from, this constraint is typically thought of resulting from some budget, with prices \(p_j\) and income \(I\): \[ p_x\cdot X + p_y \cdot Y \le I \]
As income shrinks relative to prices, our constraint tightens
Almost always, this results in a decline in agent’s utility
Notice, in the previous Chicken exercise, Barry added a constraint and it subsequently increased his payoff
Or, if we think of this in reverse: giving Barry more freedoms actually reduces his equilibrium payoffs
Finding SPNE are pretty straightforward on a game tree
In fact, given an extensive form game:
Need (plain ol’) NE?
Write down strategies for each player
Convert to Normal Form
Method: Underline
Need SPNE?
Use the game tree directly
Method: Backward Induction1
In the Entrant Game, (Out, Fight) seems like a bad prediction
If the entrant decides to go ‘In’, is it likely that the incumbent will choose to fight?
Think ‘Ultimatum game’
For now, we will assume singleton info. sets (no dashed lines)
Like too much of game theory, this method is hard to write down, but easy to display and straightforward to perform once you get the hang of it
Let’s do an example w/ the Entrant Game
How many subgames?
With common knowledge of sequential rationality, player 1 knows this about player 2
So, let’s just replace SG1 with the payoff that will be realized if player 1 plays Enter
Start with the lowest (endmost) subgames on a tree
Repeat for all ‘levels’ of the tree, working your way up to the top
There are a few exceptions to the following rule, but they are far outside the scope of this class
Therefore, I’m fine with you treating the following as a fact
Consider the following game
What about SPNE?
Start
Throughout the game, players take turns playing “end” or “continue”
Playing “end” ends the game, and both players get all of the money in the pot they are holding
Playing “continue” adds $1 to each pot and both players swap pots
The total pool of money is $200. If both pots have \(\ge \$99\) and the game continues, then both players receive $100 and the game ends
Here is the previous game in extensive form
Any Predictions?
Where would you end?
In the last stage of the game, player 2 chooses to end rather than continue to get the extra dollar
Knowing that player 2 will do this, player 1 will end the game a round early, yielding \((100, 98)\)
…but player 2 can reason to this point1, so they will end a round before that
But what does player 1 do before this?
We continue this line of thought alllll the way back up the game ‘tree’
The result from backward induction is that player 1 ends the game on their first move
This sort of phenomenon is commonly known as unravelling
Consider the following variant on the kidnapping game - How many subgames are there?